Displaying similar documents to “Regularity properties of commutators and B M O -Triebel-Lizorkin spaces”

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

Similarity:

For 1 ≤ q ≤ α ≤ p ≤ ∞, ( L q , l p ) α is a complex Banach space which is continuously included in the Wiener amalgam space ( L q , l p ) and contains the Lebesgue space L α . We study the closure ( L q , l p ) c , 0 α in ( L q , l p ) α of the space of test functions (infinitely differentiable and with compact support in d ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space W ¹ ( ( L q , l p ) α ) (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

A characterization of Sobolev spaces via local derivatives

David Swanson (2010)

Colloquium Mathematicae

Similarity:

Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function f W k , p ( Ω ) possesses an L p derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space W k , p ( Ω ) . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

Similarity:

Let ( X , d , μ ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ . Let L be a non-negative self-adjoint operator of order m on L 2 ( X ) . Assume that the semigroup e - t L generated by L satisfies the Davies-Gaffney estimate of order m and L satisfies the Plancherel type estimate. Let H L p ( X ) be the Hardy space associated with L . We show the boundedness of Stein’s square function 𝒢 δ ( L ) arising from Bochner-Riesz means associated to L from Hardy spaces H L p ( X ) to L p ( X ) , and also study...

Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa (2014)

Journal of the European Mathematical Society

Similarity:

For 1 n < d integers and ρ > 2 , we prove that an n -dimensional Ahlfors-David regular measure μ in d is uniformly n -rectifiable if and only if the ρ -variation for the Riesz transform with respect to μ is a bounded operator in L 2 ( μ ) . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L 2 ( μ ) boundedness of the Riesz transform to the uniform rectifiability of μ .

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang (2016)

Czechoslovak Mathematical Journal

Similarity:

Let 1 = - Δ + V be a Schrödinger operator and let 2 = ( - Δ ) 2 + V 2 be a Schrödinger type operator on n ( n 5 ) , where V 0 is a nonnegative potential belonging to certain reverse Hölder class B s for s n / 2 . The Hardy type space H 2 1 is defined in terms of the maximal function with respect to the semigroup { e - t 2 } and it is identical to the Hardy space H 1 1 established by Dziubański and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator b = b f - ( b f ) generated by the Riesz transform = 2 2 - 1 / 2 , where b BMO θ ( ρ ) , which is larger...

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

Shuichi Sato (2019)

Czechoslovak Mathematical Journal

Similarity:

We consider Littlewood-Paley functions associated with a non-isotropic dilation group on n . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted L p spaces, 1 < p < , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

Second order elliptic operators with complex bounded measurable coefficients in  L p , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let  L be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with L , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in  L p , Sobolev, and some new Hardy spaces naturally associated to  L . First, we show...

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

Journal of the European Mathematical Society

Similarity:

We define a new function space B , which contains in particular BMO, BV, and W 1 / p , p , 1 < p < . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving L p norms of integer-valued functions in B . We introduce a significant closed subspace, B 0 , of B , containing in particular VMO and W 1 / p , p , 1 p < . The above mentioned estimates imply in particular that integer-valued functions belonging to B 0 are necessarily constant. This framework provides a “common roof”...

Boundedness of para-product operators on spaces of homogeneous type

Yayuan Xiao (2017)

Czechoslovak Mathematical Journal

Similarity:

We obtain the boundedness of Calderón-Zygmund singular integral operators T of non-convolution type on Hardy spaces H p ( 𝒳 ) for 1 / ( 1 + ϵ ) < p 1 , where 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss (1971), and ϵ is the regularity exponent of the kernel of the singular integral operator T . Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast...

One-sided discrete square function

A. de la Torre, J. L. Torrea (2003)

Studia Mathematica

Similarity:

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average A f ( x ) = 2 - n x x + 2 f . The square function is defined as S f ( x ) = ( n = - | A f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 . The local version of this operator, namely the operator S f ( x ) = ( n = - 0 | A f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 , is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps L p into itself (p > 1) and L into BMO. We prove that the operator S not only maps L into BMO but it also maps BMO into BMO. We also prove that the L p boundedness...

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

Similarity:

We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

For a function f L l o c p ( ) the notion of p-mean variation of order 1, p ( f , ) is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space W 1 , p ( ) in terms of p ( f , ) is directly related to the characterisation of W 1 , p ( ) by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

The Hausdorff operators on the real Hardy spaces H p ( )

Yuichi Kanjin (2001)

Studia Mathematica

Similarity:

We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space H p ( ) , 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on H p ( ) , 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of H p ( ) .